Problem: Determine how many solutions exist for the system of equations. ${3x+y = 5}$ ${9x+3y = -6}$
Answer: Convert both equations to slope-intercept form: ${3x+y = 5}$ $3x{-3x} + y = 5{-3x}$ $y = 5-3x$ ${y = -3x+5}$ ${9x+3y = -6}$ $9x{-9x} + 3y = -6{-9x}$ $3y = -6-9x$ $y = -2-3x$ ${y = -3x-2}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = -3x+5}$ ${y = -3x-2}$ Both equations have the same slope with different y-intercepts. This means the equations are parallel. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Parallel lines never intersect, thus there are NO SOLUTIONS.